City Hall sees players competing to become Mayor of New York City. They do this by attempting to be the most successful at both bringing people into the city as well as campaigning for the citizens' approval. Whoever best balances these two goals will win the election.
There are seven offices within City Hall. These offices deal with a different aspect of building the city or campaigning, such as the Tax Assessor, Surveyor, or Zoning Board. In a round, each player will get to activate one of these offices. However, just because you activate an office doesn't mean you will get to use it. The other players will have an opportunity to use their influence to steal control of the office away from you. Keeping it will require countering with your own influence. However, you can instead let another player control that office this round and add their influence to your own, giving you a leg up on controlling things later on.